Problem: Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 14 centimeters about its vertical line of symmetry. Express your answer in terms of $\pi$.
Answer: [asy]
size(100);
draw((-5,-.2)--(-3,-.2)--(-3,1.8)--(-5,1.8)--cycle); label("14",((-3,1.8)--(-5,1.8)),N); label("14",((-5,-.2)--(-5,1.8)),W);
draw((-2.5,.9)--(-1.5,.9),EndArrow);
import solids; import three; currentprojection = orthographic(5,0,2);
revolution c = cylinder((0,0,0), 1, 2);
draw((0,-1,2)--(0,1,2)); label("14",((0,-1,2)--(0,1,2)),N); label("14",(0,1,1),E);
draw(c,black);
[/asy]

Rotating the square about its vertical line of symmetry creates a right circular cylinder with diameter 14 and height 14.  Thus, the cylinder has radius $14/2=7$ and volume $\pi(7^2)(14)=\pi(50-1)(14)=\pi(700-14)=\boxed{686\pi}$.